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Theorem riotasv2s 36595
Description: The value of description binder 𝐷 for a single-valued class expression 𝐶(𝑦) (as in e.g. reusv2 5270) in the form of a substitution instance. Special case of riota2f 7152. (Contributed by NM, 3-Mar-2013.) (Proof shortened by Mario Carneiro, 6-Dec-2016.)
Hypothesis
Ref Expression
riotasv2s.2 𝐷 = (𝑥𝐴𝑦𝐵 (𝜑𝑥 = 𝐶))
Assertion
Ref Expression
riotasv2s ((𝐴𝑉𝐷𝐴 ∧ (𝐸𝐵[𝐸 / 𝑦]𝜑)) → 𝐷 = 𝐸 / 𝑦𝐶)
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝑥,𝐶   𝑥,𝐸,𝑦   𝜑,𝑥
Allowed substitution hints:   𝜑(𝑦)   𝐶(𝑦)   𝐷(𝑥,𝑦)   𝑉(𝑥,𝑦)

Proof of Theorem riotasv2s
StepHypRef Expression
1 3simpc 1151 . 2 ((𝐴𝑉𝐷𝐴 ∧ (𝐸𝐵[𝐸 / 𝑦]𝜑)) → (𝐷𝐴 ∧ (𝐸𝐵[𝐸 / 𝑦]𝜑)))
2 simp1 1137 . 2 ((𝐴𝑉𝐷𝐴 ∧ (𝐸𝐵[𝐸 / 𝑦]𝜑)) → 𝐴𝑉)
3 riotasv2s.2 . . . . . 6 𝐷 = (𝑥𝐴𝑦𝐵 (𝜑𝑥 = 𝐶))
4 nfra1 3131 . . . . . . 7 𝑦𝑦𝐵 (𝜑𝑥 = 𝐶)
5 nfcv 2899 . . . . . . 7 𝑦𝐴
64, 5nfriota 7140 . . . . . 6 𝑦(𝑥𝐴𝑦𝐵 (𝜑𝑥 = 𝐶))
73, 6nfcxfr 2897 . . . . 5 𝑦𝐷
87nfel1 2915 . . . 4 𝑦 𝐷𝐴
9 nfv 1921 . . . . 5 𝑦 𝐸𝐵
10 nfsbc1v 3700 . . . . 5 𝑦[𝐸 / 𝑦]𝜑
119, 10nfan 1906 . . . 4 𝑦(𝐸𝐵[𝐸 / 𝑦]𝜑)
128, 11nfan 1906 . . 3 𝑦(𝐷𝐴 ∧ (𝐸𝐵[𝐸 / 𝑦]𝜑))
13 nfcsb1v 3814 . . . 4 𝑦𝐸 / 𝑦𝐶
1413a1i 11 . . 3 ((𝐷𝐴 ∧ (𝐸𝐵[𝐸 / 𝑦]𝜑)) → 𝑦𝐸 / 𝑦𝐶)
1510a1i 11 . . 3 ((𝐷𝐴 ∧ (𝐸𝐵[𝐸 / 𝑦]𝜑)) → Ⅎ𝑦[𝐸 / 𝑦]𝜑)
163a1i 11 . . 3 ((𝐷𝐴 ∧ (𝐸𝐵[𝐸 / 𝑦]𝜑)) → 𝐷 = (𝑥𝐴𝑦𝐵 (𝜑𝑥 = 𝐶)))
17 sbceq1a 3691 . . . 4 (𝑦 = 𝐸 → (𝜑[𝐸 / 𝑦]𝜑))
1817adantl 485 . . 3 (((𝐷𝐴 ∧ (𝐸𝐵[𝐸 / 𝑦]𝜑)) ∧ 𝑦 = 𝐸) → (𝜑[𝐸 / 𝑦]𝜑))
19 csbeq1a 3804 . . . 4 (𝑦 = 𝐸𝐶 = 𝐸 / 𝑦𝐶)
2019adantl 485 . . 3 (((𝐷𝐴 ∧ (𝐸𝐵[𝐸 / 𝑦]𝜑)) ∧ 𝑦 = 𝐸) → 𝐶 = 𝐸 / 𝑦𝐶)
21 simpl 486 . . 3 ((𝐷𝐴 ∧ (𝐸𝐵[𝐸 / 𝑦]𝜑)) → 𝐷𝐴)
22 simprl 771 . . 3 ((𝐷𝐴 ∧ (𝐸𝐵[𝐸 / 𝑦]𝜑)) → 𝐸𝐵)
23 simprr 773 . . 3 ((𝐷𝐴 ∧ (𝐸𝐵[𝐸 / 𝑦]𝜑)) → [𝐸 / 𝑦]𝜑)
2412, 14, 15, 16, 18, 20, 21, 22, 23riotasv2d 36594 . 2 (((𝐷𝐴 ∧ (𝐸𝐵[𝐸 / 𝑦]𝜑)) ∧ 𝐴𝑉) → 𝐷 = 𝐸 / 𝑦𝐶)
251, 2, 24syl2anc 587 1 ((𝐴𝑉𝐷𝐴 ∧ (𝐸𝐵[𝐸 / 𝑦]𝜑)) → 𝐷 = 𝐸 / 𝑦𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1088   = wceq 1542  wnf 1790  wcel 2114  wnfc 2879  wral 3053  [wsbc 3680  csb 3790  crio 7126
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-sep 5167  ax-nul 5174  ax-pow 5232  ax-pr 5296  ax-un 7479  ax-riotaBAD 36590
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ral 3058  df-rex 3059  df-reu 3060  df-rab 3062  df-v 3400  df-sbc 3681  df-csb 3791  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-nul 4212  df-if 4415  df-pw 4490  df-sn 4517  df-pr 4519  df-op 4523  df-uni 4797  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5429  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-iota 6297  df-fun 6341  df-fv 6347  df-riota 7127  df-undef 7968
This theorem is referenced by: (None)
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